Copied to
clipboard

G = M4(2).D10order 320 = 26·5

12nd non-split extension by M4(2) of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).12D10, C8⋊C222D5, (C5×D4).11D4, C4.178(D4×D5), C52C8.46D4, (C5×Q8).11D4, D4⋊D104C2, C4○D4.24D10, (C2×D4).79D10, C20.195(C2×D4), C20.D49C2, C55(D4.4D4), D4.4(C5⋊D4), D4.Dic55C2, Q8.4(C5⋊D4), C20.53D49C2, (C2×C20).14C23, C20.46D410C2, C10.124(C4⋊D4), (C2×D20).133C22, (D4×C10).104C22, C4.Dic5.24C22, C2.30(Dic5⋊D4), C22.13(D42D5), (C5×M4(2)).22C22, (C2×D4⋊D5)⋊22C2, (C5×C8⋊C22)⋊6C2, C4.51(C2×C5⋊D4), (C2×C4).14(C22×D5), (C2×C10).36(C4○D4), (C5×C4○D4).12C22, (C2×C52C8).170C22, SmallGroup(320,826)

Series: Derived Chief Lower central Upper central

C1C2×C20 — M4(2).D10
C1C5C10C20C2×C20C2×D20D4⋊D10 — M4(2).D10
C5C10C2×C20 — M4(2).D10
C1C2C2×C4C8⋊C22

Generators and relations for M4(2).D10
 G = < a,b,c,d | a8=b2=c10=1, d2=a6, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 446 in 108 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C20, C20, D10, C2×C10, C2×C10, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C8⋊C22, C52C8, C52C8, C40, D20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×C10, D4.4D4, C2×C52C8, C2×C52C8, C4.Dic5, C4.Dic5, D4⋊D5, Q8⋊D5, C5×M4(2), C5×D8, C5×SD16, C2×D20, D4×C10, C5×C4○D4, C20.53D4, C20.46D4, C20.D4, C2×D4⋊D5, D4.Dic5, D4⋊D10, C5×C8⋊C22, M4(2).D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, C22×D5, D4.4D4, D4×D5, D42D5, C2×C5⋊D4, Dic5⋊D4, M4(2).D10

Smallest permutation representation of M4(2).D10
On 80 points
Generators in S80
(1 17 40 21 63 41 59 78)(2 79 60 42 64 22 31 18)(3 19 32 23 65 43 51 80)(4 71 52 44 66 24 33 20)(5 11 34 25 67 45 53 72)(6 73 54 46 68 26 35 12)(7 13 36 27 69 47 55 74)(8 75 56 48 70 28 37 14)(9 15 38 29 61 49 57 76)(10 77 58 50 62 30 39 16)
(1 63)(3 65)(5 67)(7 69)(9 61)(12 46)(14 48)(16 50)(18 42)(20 44)(22 79)(24 71)(26 73)(28 75)(30 77)(32 51)(34 53)(36 55)(38 57)(40 59)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)
(1 10 59 39 63 62 40 58)(2 57 31 61 64 38 60 9)(3 8 51 37 65 70 32 56)(4 55 33 69 66 36 52 7)(5 6 53 35 67 68 34 54)(11 12 72 26 45 46 25 73)(13 20 74 24 47 44 27 71)(14 80 28 43 48 23 75 19)(15 18 76 22 49 42 29 79)(16 78 30 41 50 21 77 17)

G:=sub<Sym(80)| (1,17,40,21,63,41,59,78)(2,79,60,42,64,22,31,18)(3,19,32,23,65,43,51,80)(4,71,52,44,66,24,33,20)(5,11,34,25,67,45,53,72)(6,73,54,46,68,26,35,12)(7,13,36,27,69,47,55,74)(8,75,56,48,70,28,37,14)(9,15,38,29,61,49,57,76)(10,77,58,50,62,30,39,16), (1,63)(3,65)(5,67)(7,69)(9,61)(12,46)(14,48)(16,50)(18,42)(20,44)(22,79)(24,71)(26,73)(28,75)(30,77)(32,51)(34,53)(36,55)(38,57)(40,59), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,10,59,39,63,62,40,58)(2,57,31,61,64,38,60,9)(3,8,51,37,65,70,32,56)(4,55,33,69,66,36,52,7)(5,6,53,35,67,68,34,54)(11,12,72,26,45,46,25,73)(13,20,74,24,47,44,27,71)(14,80,28,43,48,23,75,19)(15,18,76,22,49,42,29,79)(16,78,30,41,50,21,77,17)>;

G:=Group( (1,17,40,21,63,41,59,78)(2,79,60,42,64,22,31,18)(3,19,32,23,65,43,51,80)(4,71,52,44,66,24,33,20)(5,11,34,25,67,45,53,72)(6,73,54,46,68,26,35,12)(7,13,36,27,69,47,55,74)(8,75,56,48,70,28,37,14)(9,15,38,29,61,49,57,76)(10,77,58,50,62,30,39,16), (1,63)(3,65)(5,67)(7,69)(9,61)(12,46)(14,48)(16,50)(18,42)(20,44)(22,79)(24,71)(26,73)(28,75)(30,77)(32,51)(34,53)(36,55)(38,57)(40,59), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,10,59,39,63,62,40,58)(2,57,31,61,64,38,60,9)(3,8,51,37,65,70,32,56)(4,55,33,69,66,36,52,7)(5,6,53,35,67,68,34,54)(11,12,72,26,45,46,25,73)(13,20,74,24,47,44,27,71)(14,80,28,43,48,23,75,19)(15,18,76,22,49,42,29,79)(16,78,30,41,50,21,77,17) );

G=PermutationGroup([[(1,17,40,21,63,41,59,78),(2,79,60,42,64,22,31,18),(3,19,32,23,65,43,51,80),(4,71,52,44,66,24,33,20),(5,11,34,25,67,45,53,72),(6,73,54,46,68,26,35,12),(7,13,36,27,69,47,55,74),(8,75,56,48,70,28,37,14),(9,15,38,29,61,49,57,76),(10,77,58,50,62,30,39,16)], [(1,63),(3,65),(5,67),(7,69),(9,61),(12,46),(14,48),(16,50),(18,42),(20,44),(22,79),(24,71),(26,73),(28,75),(30,77),(32,51),(34,53),(36,55),(38,57),(40,59)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,10,59,39,63,62,40,58),(2,57,31,61,64,38,60,9),(3,8,51,37,65,70,32,56),(4,55,33,69,66,36,52,7),(5,6,53,35,67,68,34,54),(11,12,72,26,45,46,25,73),(13,20,74,24,47,44,27,71),(14,80,28,43,48,23,75,19),(15,18,76,22,49,42,29,79),(16,78,30,41,50,21,77,17)]])

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C5A5B8A8B8C8D8E8F8G10A10B10C10D10E···10J20A20B20C20D20E20F40A40B40C40D
order1222224445588888881010101010···1020202020202040404040
size112484022422810102020204022448···84444888888

38 irreducible representations

dim1111111122222222224448
type+++++++++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C5⋊D4C5⋊D4D4.4D4D4×D5D42D5M4(2).D10
kernelM4(2).D10C20.53D4C20.46D4C20.D4C2×D4⋊D5D4.Dic5D4⋊D10C5×C8⋊C22C52C8C5×D4C5×Q8C8⋊C22C2×C10M4(2)C2×D4C4○D4D4Q8C5C4C22C1
# reps1111111121122222442222

Matrix representation of M4(2).D10 in GL8(𝔽41)

21332110000
156220000
37238380000
27393170000
0000228015
00002281715
0000121200
000037371211
,
10000000
01000000
00100000
00010000
000040000
000004000
00000010
00003101
,
184017240000
36233800000
12161710000
292840240000
00000010
0000342519
00001000
000033403916
,
202424200000
26213200000
0024380000
001170000
0000342519
00000010
00001000
00004033397

G:=sub<GL(8,GF(41))| [21,15,37,27,0,0,0,0,3,6,2,39,0,0,0,0,32,2,38,3,0,0,0,0,11,2,38,17,0,0,0,0,0,0,0,0,2,2,12,37,0,0,0,0,28,28,12,37,0,0,0,0,0,17,0,12,0,0,0,0,15,15,0,11],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,3,0,0,0,0,0,40,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[18,36,12,29,0,0,0,0,40,23,16,28,0,0,0,0,17,38,17,40,0,0,0,0,24,0,1,24,0,0,0,0,0,0,0,0,0,34,1,33,0,0,0,0,0,25,0,40,0,0,0,0,1,1,0,39,0,0,0,0,0,9,0,16],[20,26,0,0,0,0,0,0,24,21,0,0,0,0,0,0,24,3,24,1,0,0,0,0,20,20,38,17,0,0,0,0,0,0,0,0,34,0,1,40,0,0,0,0,25,0,0,33,0,0,0,0,1,1,0,39,0,0,0,0,9,0,0,7] >;

M4(2).D10 in GAP, Magma, Sage, TeX

M_4(2).D_{10}
% in TeX

G:=Group("M4(2).D10");
// GroupNames label

G:=SmallGroup(320,826);
// by ID

G=gap.SmallGroup(320,826);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,1123,297,136,1684,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

׿
×
𝔽